Question: What is the extraneous solution to these equations? $\dfrac{x^2 - 2}{x - 4} = \dfrac{14x - 42}{x - 4}$
Answer: Multiply both sides by $x - 4$ $ \dfrac{x^2 - 2}{x - 4} (x - 4) = \dfrac{14x - 42}{x - 4} (x - 4)$ $ x^2 - 2 = 14x - 42$ Subtract $14x - 42$ from both sides: $ x^2 - 2 - (14x - 42) = 14x - 42 - (14x - 42)$ $ x^2 - 2 - 14x + 42 = 0$ $ x^2 + 40 - 14x = 0$ Factor the expression: $ (x - 10)(x - 4) = 0$ Therefore $x = 10$ or $x = 4$ At $x = 4$ , the denominator of the original expression is 0. Since the expression is undefined at $x = 4$, it is an extraneous solution.